We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the...

We give an equivalence criterion on property A and property B for delay third order linear differential equations. We also give comparison results on properties A and B between linear and nonlinear equations, whereby we only suppose that nonlinearity has superlinear growth near infinity.

Sufficient conditions are established for the oscillation of proper solutions of the system u1'(t)=p(t)u2(σ(t)),u2'(t)=-q(t)u1(τ(t)),
where p,q:R+→R+ are locally summable functions, while τ and σ:R+→R+ are continuous and continuously differentiable functions, respectively, and limt→+∞τ(t)=+∞, limt→+∞σ(t)=+∞.

This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as t→∞.

The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation
[p(t)[(r(t)xΔ(t))Δ]γ]Δ+q(t)f(x(τ(t)))=0, t ≥ t₀,
on a time scale , where γ > 0 is a quotient of odd positive integers, and p, q, r and τ are positive right-dense continuous functions defined on . We classify the nonoscillatory solutions into certain classes Ci, i = 0,1,2,3, according to the sign of the Δ-quasi-derivatives and obtain sufficient conditions in order that Ci=∅. Also, we establish...

This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form x˙(t)=-c(t)x(t-τ(t))(*)
with positive function c(t). Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation y˙(t)=β(t)[y(t)-y(t-τ(t))]
where the function β(t) is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of...